That depends on the circumstances. If it's an actual dual, Jhin wins.
Whisper is a unique weapon in the sense that she uses not only magic, but Jhin's own chi techniques to turn people into artwork. While Cait's rifle is exceptionally crafted, Whisper will kill you headshot or not. It doesn't mater where a bullet from Whisper hits you, it'll still twist and contort that limb beyond recognition as anything human. If you combine that and the Hextech Mass Accelerator(the thing on Jhin's shoulder), you get an extremely powerful shot that will either rip your limbs off or kill you instantly, depending on where it hits. Even without the Accelerator, Whisper can still be converted into a rifle with just Jhin's cane, which allows for a quieter execution, but it lacks the lethal kick that the Accelerator would give it.
If Caitlyn knows before hand and is able to get in position before Jhin does, or simply catches him off guard, then she would most likely win.
While Cait's rifle lacks Whisper's lethal kick, it's still a weapon. It may not turn your arm into a golden tree branch with flowers blooming from your fingertips when the bullet hits you, but it'll still hurt. There is also no coming back from a direct hit to your head. While I'm not sure what the properties of Jhin's mask are, I'm pretty sure being bulletproof is not one of them.
If Jhin gets the upper hand on Caitlyn, then he definitely wins.
For the reasons I stated above, there really is no surviving a shot from Whisper, especially if you're not expecting to get shot.
Jhin also has a line for Caitlyn, something along the lines of, "You shoot to kill. You'd better hit the heart." Implying that Jhin prefers to hit his targets in the heart/chest rather than their head. Unless he's feeling particularly sadistic/killing for recreational purposes instead of assassinating someone for the cabal. Then he'll just torture you. Caitlyn however(and most of the female cast to be honest) seem to enjoy flaunting their weak points, so Jhin may as well take advantage of that fact.